Theorem crnggrpd 20180

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20179	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20175	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2113  Grpcgrp 18861  CRingccrg 20167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-iota 6446  df-fv 6498  df-ov 7359  df-ring 20168  df-cring 20169

This theorem is referenced by:  fermltlchr  21482  psdmul  22107  psd1  22108  psdpw  22111  ply1chr  22248  ply1fermltlchr  22254  elrgspnsubrunlem1  33278  elrgspnsubrunlem2  33279  erlbr2d  33295  rlocaddval  33299  rloccring  33301  rloc0g  33302  rlocf1  33304  fracerl  33337  gsumind  33375  ressply1evls1  33595  evl1deg1  33606  evl1deg2  33607  evl1deg3  33608  ply1dg1rt  33610  vr1nz  33623  esplyfvn  33682  irngss  33793  extdgfialglem1  33798  irredminply  33822  algextdeglem4  33826  algextdeglem5  33827  aks6d1c1p3  42303  aks6d1c2lem4  42320  aks6d1c6lem2  42364  aks5lem2  42380  evlsbagval  42754  selvvvval  42770  evlselv  42772  selvadd  42773  prjcrv0  42818